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# Unit Cells in a Crystal Lattice

List of Contents

### Unit Cell:

A unit cell is the smallest repeated portion of a crystal lattice that shows the three-dimensional pattern of the entire crystal. The unit cells are used to define the lengths of various crystallographic axes. When arrays of atoms or molecules are laid out in a space lattice we define a group of such atoms as the unit cell.  This unit cell contains all the necessary points on the lattice that can be translated to repeat itself in an infinite array.  In other words, the unit cell defines the basic building blocks of the crystal, and the entire crystal is made up of repeatedly translated unit cells. In defining a unit cell for a crystal the choice is somewhat arbitrary.  But, the best choice is one where:

1. The edges of the unit cell should coincide with the symmetry of the lattice.
2. The edges of the unit cell should be related by the symmetry of the lattice.
3. The smallest possible cell that contains all elements should be chosen.

For example, in the 2-dimensional lattice shown here there are 6 possible choices to define the unit cell, labeled a through f.  The lattice has 2-fold rotational symmetry about an axis perpendicular to the page.  Since the lattice itself does not have 3-fold or 6-fold rotational symmetry, choices a and b would not be wise choices for the unit cell.  Choice f can be eliminated because it is really just half of cell b.  The edges of  ‘c’ and ‘e’ are not coincident or parallel to any 2-fold axes that lie in the plane of the page.  Thus our best choice would cell ‘d’.

Once we have chosen a unit cell for the crystal, then it can be oriented on the crystallographic axes to define the angles between the axes and to define the axial lengths.  This will allow us to define directions within the crystal that become important when we realize that many properties of crystals depend on the direction in the crystal.  Properties that depend on the direction in the crystal are called vectorial properties.

Another important point is that the relative lengths of the crystallographic axes, or unit cell edges, can be determined from measurements of the angles between crystal faces.

### Unit Cells of a Cubic Crystal:

There are three basic types of unit cells in all crystal forms.

• Simple Cubic Unit Cell
• Body-Centered Cubic Unit Cell
• Face Centered Cubic Unit Cell Unit cells are the smallest repeated portions of a crystal lattice.
1. Simple Cubic Unit Cells: In ”simple cubic unit cell” atoms are present only in the corners of the cube. There are total of eight atoms in the cube. Each atom is bonded with other atoms in three dimensions. The crystal lattice further develops from the cubic unit cell thereby forming a cubic lattice. This is one of the most common and simplest shapes found in crystals and minerals. Simple cubic unit cell is shown in the above figure.
2. Body-Centered Cubic Unit Cell: The body-centered cubic system (cI) has one lattice point in the center of the unit cell in addition to the eight corner points. It has a net total of 2 lattice points per unit cell (18 × 8 + 1).
3. Face Centered Cubic Unit Cell: The face-centered cubic system (cF) has lattice points on the faces of the cube, that each gives exactly one half contribution, in addition to the corner lattice points, giving a total of 4 lattice points per unit cell (18 × 8 from the corners plus 12 × 6 from the faces). Each sphere in a cF lattice has coordination number 12. The coordination number is the number of nearest neighbours of a central atom in the structure.The face-centered cubic system is closely related to the hexagonal close-packed (hcp) system, where two systems differ only in the relative placements of their hexagonal layers. The  plane of a face-centered cubic system is a hexagonal grid.

### Unit Cells of Rhombohedral and Hexagonal Crystal Lattice:

The hexagonal crystal family consists of two lattice systems: hexagonal and rhombohedral. Each lattice system consists of one Bravais lattice. ### Unit Cells of Tetragonal Crystal Lattices:

There are two types of tetragonal crystal lattices of this crystal system, one is Simple Tetragonal, and the other is Body-Centered Tetragonal Crystal Lattice. ### Unit Cells of Monoclinic Crystal Systems:

Monoclinic Crystal Systems have ”simple” and ”based-centered” crystal lattices. Unit cells of both the lattices are as under. ### Unit Cells of Orthorhombic Crystal Systems:

The orthorhombic crystal system has three types of crystal lattices, (i) Simple orthorhombic crystal Unit Cells, (ii) Body-Centered Unit Cells, (iii) Base-Centered Unit Cells, and (iv) Face-centered Unit Cells. 